What you have been told is not correct, because if one spouse becomes Muslim before the other, then the other becomes Muslim before the woman’s ‘iddah is over, then their original marriage remains valid. The woman’s ‘iddah is three menstrual periods if she menstruates, or three months if she is past menopause, or until delivery if she is pregnant. This is the view of al-Shaafa’i and Ahmad, and is the view of Maalik with regard to cases like that mentioned in the question, which is where the wife becomes Muslim before her husband. This is also indicated by many instances that are mentioned in the Sunnah.
For example: the wife of Safwaan ibn Umayyah became Muslim on the day of the Conquest of Makkah, then he became Muslim approximately one month after her, and the Prophet (peace and blessings of Allaah be upon him) did not order them to separate or tell them to make a new marriage contract. She remained with him on the basis of their original marriage. Ibn ‘Abd al-Barr (may Allaah have mercy on him) said: The fame of this hadeeth is stronger than its isnaad.
But if the second partner became Muslim after the end of the ‘iddah, in this case there is a difference of scholarly opinion. The correct view is that if they agree to go back to one another on the basis of the original marriage contract and the woman has not married someone else, that is permissible and they do not need to do a new marriage contract. This is the view favoured by Shaykh al-Islam Ibn Taymiyah and his student Ibn al-Qayyim. It is also the view regarded as most correct by Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on them all). They quoted as evidence the report narrated by Abu Dawood from Ibn ‘Abbaas (may Allaah be pleased with him), according to whom the Messenger of Allaah (peace and blessings of Allaah be upon him) returned his daughter Zaynab to her husband Abu’l-Aas on the basis of their original marriage contract. Narrated by al-Tirmidhi, 1134; Abu Dawood, 2230; Ibn Maajah, 2019; classed as saheeh by al-Albaani in Saheeh Ibn Maajah.
He became Muslim two years after the revelation of the verses of al-Mumtahanah, in which it says that Muslim woman are forbidden to mushrik men. It seems that her ‘iddah would have ended within this period, but the Prophet (peace and blessings of Allaah be upon him) still returned her to him on the basis of the original marriage contract.
The point is that they remained married on the basis of their original marriage contract and they did not need to do a new contract. And Allaah knows best.
See Zaad al-Ma’aad, 5/133-140; al-Mughni, 10/8-10; al-Sharh al-Mumti’, 10/288-291.